The First Law of Thermodynamics: $\Delta E = q + w$

The First Law of Thermodynamics states that the change in the internal energy of a system depends on:

1. The system exchanging heat with the surroundings.
2. The system either doing work on the surroundings or the surroundings doing work on the system.

Exercise:

Consider the reaction of aluminum metal with hydrochloric acid:

\[ 2 \text{Al}(s) + 6 \text{HCl}(aq) \rightarrow 2 \text{AlCl}_3(aq) + 3 \text{H}_2(g) \]

When 5.00 g of aluminum is added to an excess of hydrochloric acid, all at a constant pressure of 1 atm and a constant temperature of 298 K, the system expels 99.8 kJ of heat into the surroundings.

1. Calculate the number of moles of hydrogen gas produced. Assume that the heat is exchanged without a change in temperature.
2. Use equations from relevant models to calculate the amount of work done by the system when the hydrogen gas expands against a constant-pressure atmosphere of 1 atm at 298 K. (Use $ R = 8.314 \times 10^{-3} $ kJ mol$^{-1}$ K$^{-1}$.)
3. Use the sign conventions of heat and work, along with the First Law of Thermodynamics, to calculate the change in internal energy of the system.

The number of moles of hydrogen gas produced is approximately 0.278 mol. The work done by the system is approximately -7420.98 kJ. The change in internal energy of the system is approximately -7519.78 kJ.

To calculate the number of moles of hydrogen gas produced and the work done by the system, we need to use the First Law of Thermodynamics, which states that the change in internal energy of a system is equal to the heat absorbed or released by the system plus the work done on or by the system.

Given:

Mass of aluminum (Al) = 5.00 g

Heat expelled by the system = -99.8 kJ

Pressure (P) = 1 atm

Temperature (T) = 298 K

Gas constant (R) = 8.314 x 10³ kJ mol⁻¹ K⁻¹

First, let's calculate the moles of aluminum used:

Molar mass of Al = 26.98 g/mol

moles of Al = mass of Al / molar mass of Al = 5.00 g / 26.98 g/mol ≈ 0.185 mol

From the balanced chemical equation, we know that 2 moles of Al react to produce 3 moles of H2. Therefore, the moles of H2 produced will be:

moles of H2 = (3/2) * moles of Al = (3/2) * 0.185 mol ≈ 0.278 mol

Next, let's calculate the work done by the system using the ideal gas law:

Work = -PΔV

Since the reaction produces gas (H2), the work done can be calculated by:

Work = -P * Δn * R * T

where Δn is the change in the number of moles of gas (moles of products - moles of reactants).

Δn = (3 - 0) = 3 moles

Work = -P * Δn * R * T = -(1 atm) * (3 mol) * (8.314 x 10³ kJ mol⁻¹ K⁻¹) * (298 K)

Work ≈ -7420.98 kJ

The negative sign indicates work done by the system.

Finally, using the First Law of Thermodynamics, we can calculate the change in internal energy of the system:

ΔU = q + w

ΔU = (-99.8 kJ) + (-7420.98 kJ) = -7519.78 kJ

The negative sign indicates that the internal energy of the system has decreased.

Therefore, the number of moles of hydrogen gas produced is approximately 0.278 mol, the work done by the system is approximately -7420.98 kJ, and the change in internal energy of the system is approximately -7519.78 kJ.

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